1 %% Vector basics
   2 
   3 %% a vector is an ordered sequence of numbers
   4 x = [2 4 6 8]
   5 
   6 y = [2:2:8]
   7 z = [8:-2:2]
   8 
   9 % getting an element out of a vector
  10 y(3)
  11 
  12 % basic operations
  13 x/2
  14 2*x
  15 whos
  16 
  17 % the number of elements of a vector
  18 length(y)
  19 % note: this is the number of elements, not the "Euclidean length"
  20 
  21 % loop over a vector
  22 for i=1:length(y)
  23     2*y(i)
  24 end
  25 
  26 % plotting vectors
  27 plot(y)
  28 hold on   % don't overwrite this
  29 plot(z, 'r')  % plot additional line in red colour
  30 
  31 whos  % look at your workspace
  32 clear all  % clear your workspace
  33 whos  % look at your (now empty) workspace
  34 close all  % close your figures
  35 
  36 %% angle between vectors
  37 x = [0 1];
  38 y = [1 1];
  39 plot([0 x(1)], [0 x(2)], 'r', [0 y(1)], [0 y(2)], 'k');
  40 % what is the angle between these two vectors?
  41 
  42 % cosine of angle between vectors
  43 cosxy = x*y' / (norm(x)*norm(y));
  44 
  45 % get the angle from cosine
  46 acosxy = acos( cosxy )
  47 
  48 % Why does is this not what we expected?
  49 % tranform radians to degrees:
  50 acosxy * (180/pi)
  51 
  52 
  53 %% plotting functions of vectors
  54 
  55 x = [-5:0.1:5]
  56 
  57 % square every element of x
  58 y = x.^2
  59 plot(y)
  60 
  61 % look at the x-axis of the plot - what's wrong?
  62 % this is better
  63 plot(x, y);
  64 plot(x, y, 'x');
  65 
  66 % a few more things
  67 hold on
  68 plot(x, (x-1).^2, 'r')
  69 hold on
  70 plot(x, x.^2+5, 'k')
  71 plot(x, 2*(x-1).^2, 'g')
  72 
  73 close all
  74 
  75 % plot sine and cosine
  76 sinx = sin(x);
  77 cosx = cos(x);
  78 
  79 plot(x, sinx, 'k', x, cosx, 'r')
  80 
  81 % correlation
  82 corr(sinx, cosx)
  83 % produces lots of NaNs because dimension not appropriate, better:
  84 corr(sinx', cosx')
  85 % sine and cosine are orthogonal - why is this correlation not exactly 0?
  86 
  87 %% Vector multiplication
  88 % we can start afresh from here
  89 
  90 % Define two parallel vectors:
  91 x = [1 0 1 0]
  92 y = [2 0 2 0]
  93 
  94 % Multiply vectors
  95 x*y  % doesn't work...
  96 
  97 % we have to "transpose" the vector on the right
  98 x*y'
  99 
 100 % transposing turns row (horizontal) vectors into column (vertical) vectors and vice versa
 101 % this is because vectors are special types of matrices - we'll come to this later
 102 x'
 103 
 104 % now try
 105 x'*y
 106 
 107 % it really matters what you are doing! a times b is not always b times a!
 108 
 109 %% Functions for two vectors
 110 
 111 % correlation between two vectors
 112 % try
 113 corr(x, y)
 114 % and you'll get a strange result
 115 % the input should be column vectors, so do it like this
 116 
 117 corr(x', y')
 118 
 119 % ttest
 120 ttest(x', y')
 121 % produces a suspiciously significant result...
 122 % check how the ttest works:
 123 help ttest
 124 
 125 % this produces the p-value
 126 [h, p] = ttest(x', y')
 127 
 128 % if you want the t-value:
 129 [h, p, ci, stats] = ttest(x', y')
 130 stats.tstat  % "stats" is a structure, with multiple elements
 131 
 132 % for simple measures
 133 sum(x)
 134 mean(x)
 135 std(x)
 136 
 137 
 138 %% Matrices
 139 
 140 % Define a matrix
 141 x = [1 1 1; 2 2 2]
 142 
 143 % Getting elements out of matrices
 144 x(1,2)
 145 x(:,1)
 146 x(2,:)
 147 
 148 % Getting matrix dimensions
 149 size(x) % returns number of rows and columns, respectively
 150 [m,n] = size(x)
 151 
 152 %% Matrices c'd
 153 % Matrix transpose
 154 x
 155 y = x'
 156 
 157 % visualise a matrix
 158 imagesc(x)
 159 imagesc(y)
 160 
 161 % adding/multiplying values etc.
 162 x + 2
 163 2 * x
 164 x.^2
 165 
 166 % Define another matrix
 167 z = [1 2 3; 4 5 6]
 168 
 169 % Subtract z from x
 170 x - z
 171 
 172 % Add z to x
 173 x + z
 174 
 175 % !!! Elementwise multiplication (NOT "matrix multiplication": "*")
 176 x .* z
 177 
 178 % Try... (matrix multiplication)
 179 x * z
 180 % doesn't work because dimensions don't match (we'll get to this later)
 181 
 182 % Elementwise division (NOT "matrix division/inversion": "/")
 183 x ./ z
 184 
 185 % Try... (matrix inversion)
 186 x / z
 187 
 188 %% Multiplying matrix and vector
 189 % Define a vector
 190 v = [1 1 1]'
 191 
 192 % Mutliply matrix x with vector v (and obtain sum within each row of x)
 193 x
 194 v
 195 x * v
 196 
 197 % matrix and vector dimensions must agree
 198 v * x % does not work
 199 
 200 %% interpretation as sum of column vectors
 201 % x * v results in the sum of column vectors in x
 202 x * v
 203 % produces the same result as
 204 1*x(:,1) + 1*x(:,2) + 1*x(:,3)
 205 
 206 %% Matrix multiplication
 207 % Multiply two matrices
 208 x * y
 209 
 210 % Multiply two matrices
 211 y * x
 212 
 213 % !!! Note: x*y is not necessarily y*x!!!
 214 
 215 % Define new matrix
 216 z = [1 1; 2 2]
 217 
 218 % Works:
 219 y*z
 220 z*x
 221 
 222 % Doesn't work:
 223 z*y
 224 x*z
 225 
 226 %% Make your own matrices
 227 
 228 % Identity matrix
 229 id = eye(3)
 230 
 231 % Example:
 232 x*id
 233 
 234 id*x'
 235 
 236 % diagonal matrix
 237 di = eye(3)
 238 di(1,1) = 1
 239 di(2,2) = 2
 240 di(3,3) = 3
 241 
 242 % Multiplies columns by a factor:
 243 x*di
 244 
 245 % Multiplies rows by a factor
 246 di*y
 247 
 248 %% Some useful functions for matrices
 249 
 250 % sum/average across columns of a matrix
 251 % note: most Matlab function operate column-by-column
 252 sum( x )
 253 mean( x )
 254 
 255 % sum across rows of a matrix
 256 sum( x' )
 257 mean( x' )
 258 
 259 % get the diagonal of a (square) matrix
 260 mydiag = diag(di)
 261 
 262 % dimensions of a matrix, size()
 263 [m,n] = size(x)
 264 
 265 % dimension of a vector
 266 diaglen = length(mydiag)
 267 
 268 % create a matrix/vector with ones
 269 ones(10)   % creates 10x10 matrix
 270 ones(10,1) % creates 10x1 column vector
 271 
 272 % create a matrix/vector with zeros
 273 zeros(10)   % creates 10x10 matrix
 274 zeros(1,10) % creates 1x10 row vector
 275 
 276 % create matrix/vector with standard uniform random numbers in interval 0-1
 277 rand(3)
 278 
 279 % create matrix/vector with standard normal random numbers in interval 0-1
 280 randn(3)
 281 
 282 % correlation among columns of a matrix
 283 R = rand(5,3)
 284 % correlation among columns
 285 corr(R)
 286 % correlation among rows
 287 corr(R')
 288 % visualise a matrix
 289 imagesc( corr(R') )
 290 colorbar
 291 
 292 
 293 % "reshaping" a matrix or a vector
 294 x = 1:10
 295 reshape(x,5,2)
 296 reshape(x,2,5)
 297 
 298 % Kronecker product ("concatenates" matrices/vectors)
 299 kron(id,y)
 300 
 301 %% EMEG example
 302 load EMEGdata.mat
 303 whos
 304 data
 305 data.evoked
 306 data.evoked.epochs
 307 
 308 % get only EEG data
 309 mydat = data.evoked.epochs(307:376,:)
 310 plot( mydat' )
 311 
 312 % find bad channel
 313 max(max(mydat))
 314 max(max(mydat'))
 315 [a,b] =max(max(mydat'))
 316 
 317 % remove bad channel
 318 mydat(63,:) = []
 319 whos
 320 
 321 plot( mydat' )
 322 
 323 [m,n] = size( mydat );
 324 
 325 % create "average reference operator"
 326 avg_op = (eye(m)-ones(m)/m);
 327 
 328 % this subtracts the mean across electrodes at each sample (column)
 329 mydatref = (eye(69)-ones(69)/69)*mydat;
 330 
 331 % this should look different:
 332 plot( mydatref' )
 333 
 334 % a different way of plotting 2D data
 335 imagesc(mydatref)
 336 colorbar
 337 
 338 % plot correlation matrix among electrodes
 339 imagesc(corr(mydatref'))
 340 colorbar